Question

The diameter if a cylinder is measured using vernier calipers with no zero error. It is found that the zero of the vernier scale lies between 5.10 cm and 5.15 cm of the main scale. The vernier scale has 50 divisions equivalent to 2.45 cm. The 24th division of the vernier scale exactly coincides with one of the main scale divisions. The diameter of the cylinder is :
A: 5.112 cm
B: 5.124 cm
C: 5.136 cm
D: 5.148 cm

Answer 1

Hint: Vernier is a device for measuring small lengths accurately, correct upto a fraction of a millimeter. The vernier calipers consist of a main scale and a sub scale called vernier scale. For this first find the least count of the vernier calipers. Where least count is the measure of ratio of one main scale division to number of divisions on the vernier. Then find the diameter of the given cylinder.
Formula used:
Diameter = Main scale reading + (Least count $\times$ Vernier Scale Reading) + Zero error

Complete answer:
Least count of a vernier is given by,
$L.C={\displaystyle \frac{1M.S.D}{n}}$
where, M.S.D is the main scale division.
 n is the total number of divisions on vernier scale
One main scale division, = 0.05cm
$n=50$
Thus least count is given by,
$L.C={\displaystyle \frac{0.05}{50}}=0.001cm$
Main scale reading is between the measure 5.10 and 5.15.
Hence,
Diameter = Main scale reading + (Least count $\times$ Vernier Scale Reading) + Zero error
Diameter = 5.10+24$\times 0.001$ = 5.124cm

Thus option(B) is correct.

Additional information:
 Uses of vernier:
(1) To measure length of small objects.
(2) To measure the internal diameter of the beaker.
(3) To determine surface area and volume of sphere.

Note:
Vernier is a device for measuring small lengths accurately, correct upto a fraction of a millimeter. On bringing the jaws of vernier calipers in contact with each other, sometimes the zero of the vernier scale may not coincide with the zero of the main scale. Then the vernier calipers are said to possess zero error. In that case we have to add zero error with the measured value.